Trigonometría Ejemplos

${(2 (\cos (72) + i \sin (72)))}^{5}$

Step 1

Simplifica los términos.

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Simplifica cada término.

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Evalúa $\cos (72)$ .

$y = {(2 (0.30901699 + i \sin (72)))}^{5}$

Evalúa $\sin (72)$ .

$y = {(2 (0.30901699 + i \cdot 0.95105651))}^{5}$

Mueve $0.95105651$ a la izquierda de $i$ .

$y = {(2 (0.30901699 + 0.95105651 i))}^{5}$

Simplifica mediante la multiplicación.

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Aplica la propiedad distributiva.

$y = {(2 \cdot 0.30901699 + 2 (0.95105651 i))}^{5}$

Multiplica.

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Multiplica $2$ por $0.30901699$ .

$y = {(0.61803398 + 2 (0.95105651 i))}^{5}$

Multiplica $0.95105651$ por $2$ .

$y = {(0.61803398 + 1.90211303 i)}^{5}$

Step 2

Usa el teorema del binomio.

$y = {0.61803398}^{5} + 5 \cdot {0.61803398}^{4} (1.90211303 i) + 10 \cdot {0.61803398}^{3} {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Step 3

Simplifica los términos.

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Simplifica cada término.

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Eleva $0.61803398$ a la potencia de $5$ .

$y = 0.09016994 + 5 \cdot {0.61803398}^{4} (1.90211303 i) + 10 \cdot {0.61803398}^{3} {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Eleva $0.61803398$ a la potencia de $4$ .

$y = 0.09016994 + 5 \cdot 0.14589803 (1.90211303 i) + 10 \cdot {0.61803398}^{3} {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $5$ por $0.14589803$ .

$y = 0.09016994 + 0.72949016 (1.90211303 i) + 10 \cdot {0.61803398}^{3} {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $1.90211303$ por $0.72949016$ .

$y = 0.09016994 + 1.38757275 i + 10 \cdot {0.61803398}^{3} {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Eleva $0.61803398$ a la potencia de $3$ .

$y = 0.09016994 + 1.38757275 i + 10 \cdot 0.23606797 {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $10$ por $0.23606797$ .

$y = 0.09016994 + 1.38757275 i + 2.36067977 {(1.90211303 i)}^{2} + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Aplica la regla del producto a $1.90211303 i$ .

$y = 0.09016994 + 1.38757275 i + 2.36067977 ({1.90211303}^{2} i^{2}) + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Eleva $1.90211303$ a la potencia de $2$ .

$y = 0.09016994 + 1.38757275 i + 2.36067977 (3.61803398 i^{2}) + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Reescribe $i^{2}$ como $- 1$ .

$y = 0.09016994 + 1.38757275 i + 2.36067977 (3.61803398 \cdot - 1) + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $2.36067977 (3.61803398 \cdot - 1)$ .

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Multiplica $3.61803398$ por $- 1$ .

$y = 0.09016994 + 1.38757275 i + 2.36067977 \cdot - 3.61803398 + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $2.36067977$ por $- 3.61803398$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 10 \cdot {0.61803398}^{2} {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Eleva $0.61803398$ a la potencia de $2$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 10 \cdot 0.38196601 {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $10$ por $0.38196601$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 {(1.90211303 i)}^{3} + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Aplica la regla del producto a $1.90211303 i$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 ({1.90211303}^{3} i^{3}) + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Eleva $1.90211303$ a la potencia de $3$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 (6.8819096 i^{3}) + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Factoriza $i^{2}$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 (6.8819096 (i^{2} \cdot i)) + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Reescribe $i^{2}$ como $- 1$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 (6.8819096 (- 1 \cdot i)) + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Reescribe $- 1 i$ como $- i$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 (6.8819096 (- i)) + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $- 1$ por $6.8819096$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 + 3.81966011 (- 6.8819096 i) + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $- 6.8819096$ por $3.81966011$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 5 \cdot 0.61803398 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Multiplica $5$ por $0.61803398$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 {(1.90211303 i)}^{4} + {(1.90211303 i)}^{5}$

Aplica la regla del producto a $1.90211303 i$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 ({1.90211303}^{4} i^{4}) + {(1.90211303 i)}^{5}$

Eleva $1.90211303$ a la potencia de $4$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 (13.09016994 i^{4}) + {(1.90211303 i)}^{5}$

Reescribe $i^{4}$ como $1$ .

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Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 (13.09016994 {(i^{2})}^{2}) + {(1.90211303 i)}^{5}$

Reescribe $i^{2}$ como $- 1$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 (13.09016994 {(- 1)}^{2}) + {(1.90211303 i)}^{5}$

Eleva $- 1$ a la potencia de $2$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 (13.09016994 \cdot 1) + {(1.90211303 i)}^{5}$

Multiplica $3.09016994 (13.09016994 \cdot 1)$ .

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Multiplica $13.09016994$ por $1$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 3.09016994 \cdot 13.09016994 + {(1.90211303 i)}^{5}$

Multiplica $3.09016994$ por $13.09016994$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + {(1.90211303 i)}^{5}$

Aplica la regla del producto a $1.90211303 i$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + {1.90211303}^{5} i^{5}$

Eleva $1.90211303$ a la potencia de $5$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + 24.89898284 i^{5}$

Factoriza $i^{4}$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + 24.89898284 (i^{4} i)$

Reescribe $i^{4}$ como $1$ .

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Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + 24.89898284 ({(i^{2})}^{2} i)$

Reescribe $i^{2}$ como $- 1$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + 24.89898284 ({(- 1)}^{2} i)$

Eleva $- 1$ a la potencia de $2$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + 24.89898284 (1 i)$

Multiplica $i$ por $1$ .

$y = 0.09016994 + 1.38757275 i - 8.54101966 - 26.2865556 i + 40.45084971 + 24.89898284 i$

Simplifica mediante la adición de términos.

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Resta $8.54101966$ de $0.09016994$ .

$y = - 8.45084971 + 1.38757275 i - 26.2865556 i + 40.45084971 + 24.89898284 i$

Suma $- 8.45084971$ y $40.45084971$ .

$y = 32 + 1.38757275 i - 26.2865556 i + 24.89898284 i$

Resta $26.2865556 i$ de $1.38757275 i$ .

$y = 32 - 24.89898284 i + 24.89898284 i$

Suma $- 24.89898284 i$ y $24.89898284 i$ .

$y = 32 + 0 i$